Question

sinx(sinx-cosx)-1=0

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Answer 1

\(\Leftrightarrow sin^2x-sinx.cosx-1=0\)

\(\Leftrightarrow-cos^2x-sinx.cosx=0\)

\(\Leftrightarrow cosx\left(cosx+sinx\right)=0\)

\(\Leftrightarrow\sqrt{2}cosx.sin\left(x+\frac{\pi}{4}\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}cosx=0\\sin\left(x+\frac{\pi}{4}\right)=0\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{2}+k\pi\\x+\frac{\pi}{4}=k\pi\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{\pi}{2}+k\pi\\x=-\frac{\pi}{4}+k\pi\end{matrix}\right.\)